Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings

We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.

From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns

Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.

Shannon (Information) Measures of Symmetry for 1D and 2D Shapes and Patterns

In this paper, informational (Shannon) measures of symmetry are introduced and analyzed for patterns built of 1D and 2D shapes. The informational measure of symmetry Hsym(G) characterizes the averaged uncertainty in the presence of symmetry elements from group G in a given pattern, whereas the Shannon-like measure of symmetry Ωsym(G) quantifies the averaged uncertainty of the appearance of shapes possessing a total of n elements of symmetry belonging to group G in a given pattern. Hsym(G1)=Ωsym(G1)=0 for the patterns built of irregular, non-symmetric shapes, where G1 is the identity element of the symmetry group. Both informational measures of symmetry are intensive parameters of the pattern and do not depend on the number of shapes, their size, and the entire area of the pattern. They are also insensitive to the long-range order (translational symmetry) inherent for the pattern. Additionally, informational measures of symmetry of fractal patterns are addressed, the mixed patterns including curves and shapes are considered, the time evolution of Shannon measures of symmetry are examined, the close-packed and dispersed 2D patterns are analyzed, and an application of the suggested measures of symmetry for the analysis of the chemical reaction is demonstrated.

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.